Finite Group Databasebeta

← #10955430

Subgroups — #10955430

Order 768.

2314 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5522C211yesmaximal, characteristic
2C3×#2346C211yesmaximal, characteristic, complemented
3#82239C211yesmaximal, characteristic, complemented
4#75982C212yesmaximal, complemented
5#75969C212yesmaximal, complemented
6#75476C214yesmaximal, complemented
7#75458C214yesmaximal, complemented
8#74156C212yesmaximal, complemented
9#74147C212yesmaximal, complemented
10#73877C211yesmaximal, characteristic
11#73863C211yesmaximal, characteristic, complemented
12#73646C212yesmaximal, complemented
13#73616C212yesmaximal, complemented
14#73454C211yesmaximal, characteristic
15#73437C211yesmaximal, characteristic, complemented
16#73413C212yesmaximal, complemented
17#73392C212yesmaximal, complemented
18#34197—31nomaximal, Sylow
19C22×#235C2211yescharacteristic
20C2×C3×#117C2211yescharacteristic
21C22×C4×Dic3C2211yescharacteristic
22#5522C4116yescomplemented
23C2×#893C2211yescharacteristic
24C2×#893C2211yescharacteristic
25C2×#887C2212yes—
26C2×#887C2211yescharacteristic
27C2×#887C2211yescharacteristic
28C2×#67⋊C4C2212yes—
29C2×#67⋊C4C2212yes—
30C2×#872C2212yes—
31C2×C42×D3C2212yes—
32C2×C4×Dic6C2212yes—
33C3×#389C2212yes—
34C3×#388C2214yes—
35C3×#382C2212yes—
36C3×#380C2211yescharacteristic
37C3×C4×C4⋊C4C2212yes—
38C3×C4×C22⋊C4C2211yescharacteristic
39C3×#376C2212yes—
40#5212C2211yescharacteristic
41C4×#217C2211yescharacteristic
42#4990C2214yes—
43#4988C2212yes—
44#4987C2214yes—
45#4984C2212yes—
46#4977C2212yes—
47#4976C2214yes—
48Dic3×C4⋊C4C2212yes—
49Dic3×C4⋊C4C2212yes—
50Dic3×C4⋊C4C2214yes—
51#4957C2214yes—
52#4957C2214yes—
53#4951C2214yes—
54#4943C2212yes—
55#4943C2212yes—
56#4942C2212yes—
57Dic3×C22⋊C4C2212yes—
58#4940C2214yes—
59#4939C2212yes—
60C4×(C2×D3)⋊C4C2214yes—
61#4936C2212yes—
62#4932C2212yes—
63#4932C2212yes—
64C4×Dic3⋊C4C2214yes—
65C4×Dic3⋊C4C2214yes—
66C4×Dic3⋊C4C2212yes—
67#4670C2214yes—
68#4669C2214yes—
69#4667C2212yes—
70#4667C2212yes—
71#4666C2212yes—
72#4666C2212yes—
73#4664C2214yes—
74#4664C2214yes—
75#4651C2212yes—
76#4651C2211yescharacteristic
77#4651C2211yescharacteristic
78#4650C2211yescharacteristic
79#4650C2211yescharacteristic
80#4650C2214yes—
81#4649C2214yes—
82#4648C2212yes—
83#4648C2214yes—
84#4647C2214yes—
85C2×C4×#41—31no—
86#2357—34no—
87#2357—32no—
88#2351—32no—
89#2346—32no—
90#2346—32no—
91#2346—31no—
92#2346D311yescharacteristic, complemented
93#2341—31no—
94#2341—31no—
95#2319—32no—
96#2319—34no—
97#2283—32no—
98#2283—31no—
99#2283—31no—
100#2283—32no—
101#2276—32no—
102C23×C3×C4C2311yescharacteristic
103C22×#67C2312yes—
104C23×Dic3C2311yescharacteristic
105C23×Dic3C2311yescharacteristic
106C2×#235—28no—
107C2×#235C2×C418yes—
108C22×D12C2311yescharacteristic
109C22×C4×D3C2312yes—
110C22×C4×D3C2312yes—
111C22×C4×D3C2312yes—
112C22×Dic6C2311yescharacteristic
113C3×#117C2×C418yes—
114C2×C3×C4⋊C4C2312yes—
115C2×C3×C4⋊C4C2314yes—
116C2×C3×C4⋊C4C2311yescharacteristic
117C2×C3×C4⋊C4C2311yescharacteristic
118C2×C3×C22⋊C4C2312yes—
119C2×C3×C22⋊C4C2312yes—
120C2×C3×C22⋊C4C2312yes—
121C2×C3×C42C2312yes—
122C2×C3×C42C2314yes—
123C2×C3×C42C2312yes—
124C2×#217C2312yes—
125C2×#217C2312yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2314yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×(C2×D3)⋊C4C2312yes—
131C2×(C2×D3)⋊C4C2312yes—
132C2×C4⋊Dic3C2312yes—
133C2×C4⋊Dic3C2314yes—
134C2×Dic3⋊C4C2314yes—
135C2×Dic3⋊C4C2314yes—
136C2×Dic3⋊C4C2312yes—
137C2×Dic3⋊C4C2312yes—
138C2×Dic3⋊C4C2312yes—
139C2×Dic3⋊C4C2312yes—
140C2×C4×Dic3C2×C418yes—
141C2×C4×Dic3C2314yes—
142C2×C4×Dic3C2314yes—
143C2×C4×Dic3C2312yes—
144C2×C4×Dic3C2314yes—
145C2×C4×Dic3C2314yes—
146C2×C4×Dic3C2311yescharacteristic
147C2×C4×Dic3C2312yes—
148C2×C4×Dic3C2312yes—
149C2×C4×Dic3C2311yescharacteristic
150C2×C4×Dic3C2311yescharacteristic
151C2×C4×Dic3C2311yescharacteristic
152#893C2×C418yes—
153#893C2×C418yes—
154#887—28no—
155#887C2×C418yes—
156#887C2×C418yes—
157#67⋊C4C2×C4116yes—
158#67⋊C4C2×C4116yes—
159#872C2×C4116yes—
160C42×D3C2×C4116yes—
161C4×Dic6—28no—
162C3×#95C2314yes—
163C3×#95C2314yes—
164C3×#95C2312yes—
165C3×#95C2311yescharacteristic
166C3×#95C2311yescharacteristic
167C3⋊#95C2314yes—
168C3⋊#95C2312yes—
169C3⋊#95C2314yes—
170C3⋊#95C2312yes—
171C3⋊#95C2314yes—
172C3⋊#95C2312yes—
173C3⋊#95C2312yes—
174C3⋊#95C2314yes—
175C3⋊#95C2311yescharacteristic
176C3⋊#95C2311yescharacteristic
177C3⋊#95C2314yes—
178C3⋊#95C2312yes—
179C3⋊#95C2312yes—
180C3⋊#95C2312yes—
181C22×#41—31no—
182C4×#41—316no—
183C2×C4×Q8—31no—
184C2×C4×Q8—31no—
185C2×C4×Q8—32no—
186C2×C4×Q8—32no—
187C2×C4×D4—31no—
188C2×C4×D4—31no—
189C2×C4×D4—32no—
190C2×C4×D4—32no—
191C2×#117—32no—
192C2×#117C2×D311yescharacteristic
193C22×C42—31no—
194C22×C42—32no—
195#390—32no—
196#390—34no—
197#390—34no—
198#390—32no—
199#389—34no—
200#389—32no—
201#389—34no—
202#389C2×D312yes—
203#388—34no—
204#388—34no—
205#388—34no—
206#388—34no—
207#388—34no—
208#388C2×D314yes—
209#383—31no—
210#383—32no—
211#383—31no—
212#382—32no—
213#382—32no—
214#382—31no—
215#382—31no—
216#382—34no—
217#382C2×D312yes—
218#381—32no—
219#381—32no—
220#381—34no—
221#381—34no—
222#380—31no—
223#380—34no—
224#380—32no—
225#380—32no—
226#380—32no—
227#380C2×D311yescharacteristic
228C4×C4⋊C4—32no—
229C4×C4⋊C4—32no—
230C4×C4⋊C4—34no—
231C4×C4⋊C4—34no—
232C4×C4⋊C4—34no—
233C4×C4⋊C4—32no—
234C4×C4⋊C4—34no—
235C4×C4⋊C4C2×D312yes—
236C4×C22⋊C4—31no—
237C4×C22⋊C4—32no—
238C4×C22⋊C4—34no—
239C4×C22⋊C4—32no—
240C4×C22⋊C4—32no—
241C4×C22⋊C4C2×D311yescharacteristic
242#376—32no—
243#376—32no—
244#376—34no—
245#376—32no—
246#376C2×D312yes—
247C24×C3C2411yescharacteristic
248C23×D3C2412yes—
249C22×C3×C4—24no—
250C22×C3×C4C22×C414yes—
251C22×C3×C4C2412yes—
252C22×C3×C4C2414yes—
253C22×C3×C4C2412yes—
254C22×C3×C4C2411yescharacteristic
255C22×C3×C4C2412yes—
256C22×C3×C4C2412yes—
257C22×C3×C4C2411yescharacteristic
258C2×#67—28no—
259C2×#67C22×C418yes—
260C22×Dic3—42no—
261C22×Dic3—41no—
262C22×Dic3—41no—
263C22×Dic3C22×C414yes—
264C22×Dic3C22×C414yes—
265C22×Dic3C2412yes—
266C22×Dic3C2412yes—
267C22×Dic3C2414yes—
268C22×Dic3C2411yescharacteristic
269C22×Dic3C2411yescharacteristic
270C22×Dic3C2411yescharacteristic
271C22×Dic3C2411yescharacteristic
272C22×Dic3C2412yes—
273#235—232no—
274C2×D12—24no—
275C2×D12C22×C414yes—
276C2×C4×D3—28no—
277C2×C4×D3—44no—
278C2×C4×D3—42no—
279C2×C4×D3—42no—
280C2×C4×D3C22×C418yes—
281C2×C4×D3C22×C418yes—
282C2×C4×D3C22×C418yes—
283C2×Dic6C22×C414yes—
284C2×Dic6#4118yes—
285C3×C4⋊C4—24no—
286C3×C4⋊C4—44no—
287C3×C4⋊C4C22×C414yes—
288C3×C4⋊C4C22×C414yes—
289C3×C22⋊C4—42no—
290C3×C22⋊C4—42no—
291C3×C22⋊C4C22×C418yes—
292C3×C42—24no—
293C3×C42—44no—
294C3×C42C22×C418yes—
295#217—42no—
296#217—42no—
297(C2×D3)⋊C4—44no—
298(C2×D3)⋊C4—44no—
299(C2×D3)⋊C4—44no—
300(C2×D3)⋊C4—44no—
301(C2×D3)⋊C4C22×C418yes—
302(C2×D3)⋊C4C22×C418yes—
303C4⋊Dic3—24no—
304C4⋊Dic3—44no—
305Dic3⋊C4—28no—
306Dic3⋊C4—28no—
307Dic3⋊C4—42no—
308Dic3⋊C4—42no—
309Dic3⋊C4C22×C418yes—
310Dic3⋊C4C22×C418yes—
311C4×Dic3—24no—
312C4×Dic3—28no—
313C4×Dic3—28no—
314C4×Dic3—44no—
315C4×Dic3—44no—
316C4×Dic3C22×C414yes—
317C4×Dic3C22×C418yes—
318C4×Dic3C22×C418yes—
319C4×Dic3C22×C414yes—
320C4×Dic3C22×C414yes—
321C4×Dic3C22×C414yes—
322C2×#41—38no—
323C2×#41—68no—
324C22×Q8—31no—
325C22×D4—32no—
326C22×D4—31no—
327C23×C4—31no—
328C23×C4—31no—
329C23×C4—32no—
330C23×C4—32no—
331C23×C4—32no—
332C23×C4C22×D311yescharacteristic
333C4×Q8—38no—
334C4×Q8—38no—
335C4×Q8—68no—
336C4×Q8—68no—
337C4×D4—38no—
338C4×D4—38no—
339C4×D4—316no—
340C4×D4—316no—
341#117—316no—
342#117C4×D318yes—
343C2×C4⋊C4—32no—
344C2×C4⋊C4—32no—
345C2×C4⋊C4—34no—
346C2×C4⋊C4—34no—
347C2×C4⋊C4—32no—
348C2×C4⋊C4—32no—
349C2×C4⋊C4—32no—
350C2×C4⋊C4—34no—
351C2×C4⋊C4C22×D312yes—
352C2×C4⋊C4C22×D314yes—
353C2×C4⋊C4C22×D311yescharacteristic
354C2×C4⋊C4C22×D311yescharacteristic
355C2×C22⋊C4—32no—
356C2×C22⋊C4—32no—
357C2×C22⋊C4—32no—
358C2×C22⋊C4—32no—
359C2×C22⋊C4—34no—
360C2×C22⋊C4—34no—
361C2×C22⋊C4—34no—
362C2×C22⋊C4—34no—
363C2×C22⋊C4C22×D312yes—
364C2×C22⋊C4C22×D312yes—
365C2×C22⋊C4C22×D312yes—
366C2×C42—38no—
367C2×C42—31no—
368C2×C42—32no—
369C2×C42—31no—
370C2×C42—32no—
371C2×C42—31no—
372C2×C42—31no—
373C2×C42—316no—
374C2×C42—34no—
375C2×C42—34no—
376C2×C42—32no—
377C2×C42—34no—
378C2×C42—34no—
379C2×C42C22×D312yes—
380C2×C42C22×D314yes—
381C2×C42C22×D312yes—
382#95—32no—
383#95—32no—
384#95—31no—
385#95—34no—
386#95—32no—
387#95—32no—
388#95—32no—
389#95—34no—
390#95—34no—
391#95—34no—
392#95—31no—
393#95—34no—
394#95—32no—
395#95—32no—
396#95C22×D312yes—
397#95C22×D314yes—
398#95C22×D311yescharacteristic
399#95C22×D314yes—
400#95C22×D311yescharacteristic
401C23×C3—22no—
402C23×C3—21no—
403C23×C3—22no—
404C23×C3—21no—
405C23×C3C23×C412yes—
406C23×C3C2511yescharacteristic
407C22×D3—24no—
408C22×D3—24no—
409C22×D3—22no—
410C22×D3—22no—
411C22×D3C23×C414yes—
412C2×C3×C4—22no—
413C2×C3×C4—24no—
414C2×C3×C4—22no—
415C2×C3×C4—22no—
416C2×C3×C4—28no—
417C2×C3×C4—22no—
418C2×C3×C4—24no—
419C2×C3×C4—22no—
420C2×C3×C4—24no—
421C2×C3×C4—22no—
422C2×C3×C4—22no—
423C2×C3×C4—22no—
424C2×C3×C4—24no—
425C2×C3×C4—22no—
426C2×C3×C4C23×C412yes—
427C2×C3×C4C23×C414yes—
428C2×C3×C4C23×C414yes—
429C2×C3×C4C23×C412yes—
430C2×C3×C4C2×#4114yes—
431#67—216no—
432C2×Dic3—22no—
433C2×Dic3—21no—
434C2×Dic3—21no—
435C2×Dic3—22no—
436C2×Dic3—21no—
437C2×Dic3—21no—
438C2×Dic3—22no—
439C2×Dic3—22no—
440C2×Dic3—24no—
441C2×Dic3—24no—
442C2×Dic3—22no—
443C2×Dic3—22no—
444C2×Dic3—22no—
445C2×Dic3—24no—
446C2×Dic3—22no—
447C2×Dic3—44no—
448C2×Dic3—44no—
449C2×Dic3C23×C412yes—
450C2×Dic3C23×C412yes—
451C2×Dic3C23×C412yes—
452C2×Dic3C23×C412yes—
453C2×Dic3C23×C414yes—
454C2×Dic3C2×#4118yes—
455D12—28no—
456C4×D3—216no—
457C4×D3—48no—
458C4×D3—48no—
459Dic6#117116yes—
460C24—32no—
461C24C23×D311yescharacteristic
462#41—632no—
463C2×Q8—34no—
464C2×Q8—38no—
465C2×D4—38no—
466C2×D4—34no—
467C2×D4—68no—
468C2×D4—64no—
469C22×C4—24no—
470C22×C4—32no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—31no—
474C22×C4—31no—
475C22×C4—34no—
476C22×C4—34no—
477C22×C4—38no—
478C22×C4—38no—
479C22×C4—38no—
480C22×C4—32no—
481C22×C4—32no—
482C22×C4—34no—
483C22×C4—68no—
484C22×C4—124no—
485C22×C4—122no—
486C22×C4—122no—
487C22×C4—122no—
488C22×C4—121no—
489C22×C4—121no—
490C22×C4C2×C4×D314yes—
491C22×C4C23×D312yes—
492C22×C4C23×D314yes—
493C22×C4C23×D312yes—
494C22×C4C23×D311yescharacteristic
495C22×C4C23×D312yes—
496C22×C4C23×D312yes—
497C22×C4C23×D311yescharacteristic
498C4⋊C4—24no—
499C4⋊C4—38no—
500C4⋊C4—38no—
501C4⋊C4—44no—
502C4⋊C4—68no—
503C4⋊C4—68no—
504C4⋊C4—64no—
505C4⋊C4—122no—
506C4⋊C4—122no—
507C4⋊C4—124no—
508C4⋊C4C2×C4×D314yes—
509C4⋊C4C2×C4×D314yes—
510C22⋊C4—38no—
511C22⋊C4—38no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—124no—
515C22⋊C4—124no—
516C22⋊C4—124no—
517C22⋊C4—124no—
518C22⋊C4—122no—
519C22⋊C4—122no—
520C22⋊C4C2×C4×D318yes—
521C42—24no—
522C42—34no—
523C42—34no—
524C42—38no—
525C42—38no—
526C42—34no—
527C42—34no—
528C42—44no—
529C42—68no—
530C42—68no—
531C42—64no—
532C42—124no—
533C42—124no—
534C42C2×C4×D318yes—
535C22×C3—22no—
536C22×C3—22no—
537C22×C3—22no—
538C22×C3—42no—
539C22×C3—42no—
540C22×C3C24×C411yescharacteristic
541C22×C3C22×#4112yes—
542C22×C3#58512yes—
543C22×C3#58511yescharacteristic
544C22×C3#58511yescharacteristic
545C2×D3—24no—
546C2×D3—24no—
547C2×D3—24no—
548C2×D3—44no—
549C2×D3—44no—
550C3×C4—22no—
551C3×C4—24no—
552C3×C4—24no—
553C3×C4—42no—
554C3×C4—42no—
555C3×C4—44no—
556C3×C4C2×#11714yes—
557Dic3—22no—
558Dic3—22no—
559Dic3—22no—
560Dic3—22no—
561Dic3—42no—
562Dic3—44no—
563Dic3—42no—
564Dic3C2×#11718yes—
565C23—22no—
566C23—22no—
567C23—21no—
568C23—21no—
569C23—34no—
570C23—64no—
571C23—64no—
572C23—62no—
573C23—62no—
574C23C22×C4×D312yes—
575C23C24×D311yescharacteristic
576Q8—316no—
577D4—616no—
578D4—68no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—28no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—34no—
594C2×C4—38no—
595C2×C4—32no—
596C2×C4—32no—
597C2×C4—32no—
598C2×C4—32no—
599C2×C4—616no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—62no—
607C2×C4—64no—
608C2×C4—62no—
609C2×C4—64no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—64no—
614C2×C4—62no—
615C2×C4—128no—
616C2×C4—128no—
617C2×C4—124no—
618C2×C4—124no—
619C2×C4C22×C4×D312yes—
620C2×C4C22×C4×D314yes—
621C2×C4C22×C4×D314yes—
622C2×C4C22×C4×D312yes—
623C2×C4D3×#4114yes—
624C2×C3—42no—
625C2×C3C22×#11711yescharacteristic
626C2×C3#342811yescharacteristic
627C2×C3#342811yescharacteristic
628C2×C3#352412yes—
629C2×C3#352412yes—
630D3—44no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—42no—
635C22—42no—
636C22—64no—
637C22—64no—
638C22—64no—
639C22—124no—
640C22—124no—
641C22C23×C4×D311yescharacteristic
642C22C2×D3×#4112yes—
643C22#596412yes—
644C22#596611yescharacteristic
645C22#596811yescharacteristic
646C4—24no—
647C4—24no—
648C4—22no—
649C4—38no—
650C4—42no—
651C4—42no—
652C4—44no—
653C4—62no—
654C4—62no—
655C4—62no—
656C4—62no—
657C4—122no—
658C4—124no—
659C4—122no—
660C4D3×#11714yes—
661C3#3419711yescharacteristic, complemented, Sylow
662C2—42no—
663C2—124no—
664C2C2×D3×#11711yescharacteristic
665C2#8583411yescharacteristic
666C2#8588511yescharacteristic
667C2#8684312yes—
668C2#8696512yes—
subgroupG/H
center, FrattiniC23C24×D3
commutatorC22×C3C24×C4
FittingC3×#2346C2
socle, cosocleC23×C3C25
prime radicalC22×#235C22
hypercenter#2346D3
hypocenterC3#34197
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2346

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×#2346C2semidirect
3. #82239C2semidirect
4. #75982C2semidirect
5. #75969C2semidirect
6. #75476C2semidirect
7. #75458C2semidirect
8. #74156C2semidirect
9. #74147C2semidirect
11. #73863C2semidirect
12. #73646C2semidirect
13. #73616C2semidirect
15. #73437C2semidirect
16. #73413C2semidirect
17. #73392C2semidirect
22. #5522C4semidirect
92. #2346D3semidirect
661. C3#34197semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#55222
C3×#23462
#822392
#759822
#759692
#754762
#754582
#741562
#741472
#738772
#738632
#736462
#736162
#734542
#734372
#734132
#733922
#341973